


Ballista

by AnnaFan



Category: The Lord of the Rings - All Media Types, The Lord of the Rings - J. R. R. Tolkien
Genre: Gen, Teitho Fanfiction Contest
Language: English
Status: Completed
Published: 2020-03-18
Updated: 2020-03-18
Packaged: 2021-02-28 20:01:25
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 3,492
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/23202892
Author URL: https://archiveofourown.org/users/AnnaFan/pseuds/AnnaFan
Summary: A half-remembered childhood experiment with a toy ballista takes on deeper significance for Erchirion of Dol Amroth during the siege of Minas Tirith.Joint first in the Teitho challenge "relationships."
Relationships: Geometrical and Numerical
Comments: 2
Kudos: 16





	Ballista

“There's no such thing as _Up-in-the-air-y-ness_.”

“Is so...”

“Isn't.”

This exchange was followed by sounds of a scuffle. Faramir hastened to the gate in the high wooden fence that surrounded the lists. The first voice, to judge by the sometimes-breaking, sometimes-not tone, was his 14-year old cousin Erchirion. The treble belonged unmistakably to nine-year-old Amrothos.

He pulled the stout wooden gate open (thinking that the way it grated on its hinges was not entirely unlike poor Erchirion).

There, in the middle of the otherwise empty lists, were his four cousins. Erchirion and Amrothos were rolling in the dirt pounding each other (Erchirion, by virtue of his size, was winning). Elphir was ignoring them, intent on scratching on a scrap of parchment with a piece of charcoal. Little Lothíriel, dabs of dirt on her face, mud a hand span deep round the hem of her dress, hair like a haystack, and a rip in her skirt, sat in the dust next to what appeared to be a rather beautifully made half-scale replica of a ballista.

Faramir strode across the close-packed dirt that made up the floor of the tilt-yard, and seized both cousins by the scruff of the neck, pulling them apart. He set them down a few paces separated from one another.

“Would you care to tell me what you were fighting about?” he said, trying simultaneously not to laugh and to imbue his voice with the cutting tone he used with young recruits who were failing to meet his exacting standards. He didn't entirely succeed with either aim.

“The ballista,” Elphir said, without looking up from his drawing. “We're trying to get to the bottom of exactly what angle to set it up at to get the stones to go the furthest distance. Erchirion reckons the flattest angle you can manage is best, because then most of the push from the bow arms goes into pushing the projectile in the direction you want it to go in, but Amrothos reckons if you have it too flat, it doesn't have enough time to get anywhere before it hits the ground.”

“Ah, I think I see,” said Faramir. “Hence _Up-in-the-air-y-ness_.”

“Yes!” shouted Amrothos, triumphantly. “See, cousin Faramir doesn't think it's a stupid idea, and he's a proper soldier.”

“I will concede that I think Amrothos is on to something. Part of the problem, though” said Elphir, “Is that the stones move too quickly to really see properly what's going on.”

“Yes, I can see that would be a problem,” Faramir replied, immediately drawn into the knotty problem the children had set themselves. He walked round the ballista in a slow circle, giving himself time to think.

“Supposing,” he said, “We used a pillow instead of a stone, and only wound the capstan up half way, so that we could watch things happen a bit more slowly?”

“We don't have a pillow,” said Erchirion, moodily. He was still cross that his adored cousin appeared to be taking sides with his intensely annoying little brother.

“We could use a sack filled with straw,” suggested Amrothos, ignoring his brother's glare.

“Right, you run over to the stable and get one. And a length of rope if you can find it – in fact, better still, some of the bunting with coloured flags on that gets used on high holidays.” The children scampered off, Lothíriel trailing in the wake of her brothers. Faramir suspected there would be hell to pay when her nursemaid saw the state she was in, but since Lothíriel clearly thought that being allowed to watch her brothers' games was a state much to be desired, he couldn't see the harm in it. (In fact, the day before it had amused him greatly to watch the nurse's scandalised face as Lothíriel demonstrated, with a tightly furled parasol, not only the high, middle and low guards of a fencer, but also managed a creditable attempt at working through the “winds from all directions”, the basic cutting strokes of the swordsman. And all at only six years old.)

When the children returned, Faramir explained his plan. The bunting was stretched out along the ground to form a crude measuring tape, Faramir having checked that the flags were indeed at roughly equal spacings. Next, Elphir and Erchirion were dispatched, each with a piece of paper and stick of charcoal, to either side of the lists. They were to record what angle the ballista was ratcheted to, where the sackcloth pillow landed, and then sketch roughly the trajectory they thought it had followed, giving their best estimate of the height reached. In this, Elphir had the advantage; his viewpoint faced the guard tower, and he would be able to estimate the height reached by matching the pillow's path against the arrow slits in the tower (for a moment he fretted about the fact that he wasn't looking straight on, and Faramir conceded this would indeed reduce the accuracy, but by way of a first approximation, it would do).

Faramir and Amrothos would set up the ballista each time, and Lothíriel was given the job of retrieving the pillow, with Faramir's help. As they went to retrieve it, they carefully counted the number of flags it had passed on the way to its final resting place.

By the time they had worked through all the possible angles the ballista allowed for (its angle was controlled by a simple ladder-rack device), repeating each attempt several times, they were ready to look at the results.

Faramir looked at the sketches the boys had done, and was pleased to see a reasonable degree of agreement. Both boys agreed that the pillow always traced out a graceful arc as it flew through the air, though all of Erchirion's sketches showed it attaining the same height, regardless of angle of launch. Elphir, judging against the tower behind, had drawn arcs of varying heights, with steeper angles giving rise to higher flights, pretty much what Faramir would have expected from his experience with archery.

But, as he looked at the sketches, something very interesting emerged. Both Erchirion and Amrothos had got a part of the puzzle. 

For angles more than half a right-angle, Erchirion seemed to be right – the steeper the launch angle, the more of the push from the ballista was “wasted” sending the missile into the air rather than forward. The steeper the angle, the higher it went, needlessly. Obviously the limiting case was to fire it straight into the air – it dropped almost directly onto the ballista (variations, Faramir assumed, being due to slight winds). The shallower the angle, the more of the push went into sending the missile forwards, and the further it went… until one reached half a right angle.

But at that point, strangely, as the angle of the ballista was dropped still further towards the ground, the pillow started dropping short of its furthest distance – and going less and less of a distance the shallower the angle became. Here, Elphir's sketches with the variation in height seemed to offer the clue. Amrothos' _up-in-the-air-y-ness_ was indeed the key to the puzzle. The pillow wasn't launched high enough to spend enough time describing its gentle arc before it hit the ground. 

Faramir frowned. Here was a puzzle indeed. It all made intuitive sense. When he had first learned the art of the bow, using a lightly strung child's bow, he had learned instinctively to compensate for long shots by raising the tip of the arrow above the target. He had likewise taught his nephews. So, intuitively, in his gut, the sketches made perfect sense. They felt right. But they ought also, particularly Elphir's, to offer a clue to the relationship between angle and distance. Faramir took great joy in the philosophical study of geometry as well as its practical applications in architecture and engineering; he felt there ought to be some pattern in the numbers that he could make out. But what?

His musings were interrupted by the arrival of Lothíriel's nursemaid, to summon them to lunch. Bridling in annoyance at the state of her charge, she swept the little girl into her arms and carried her off to make her presentable. Faramir took the boys back to the palace and handed them over to their tutor for him to inflict a scrubbing with sponge and wash cloth before lunch, then retired to his own chamber to wash himself and change his tunic. As he was about to leave to go to the family's dining room, he paused for a moment. He collected the sketches and tucked them into a battered leather folder which he normally used for maps of troop dispositions, and tucked them into his satchel. Then, stomach growling, he set off for Imrahil's table.

_Thirteen years later_

Erchirion felt a pang of grief as he entered his cousin's room. Although most of the women of the city had left, his mother had filled their town house with widows and middle-aged spinsters – dependable, stout women who could be trusted not to panic in a crisis – to assist the healers. So it was that her sons, grasping a moment's respite from resisting the siege in order to snatch a brief sleep, found themselves in the Steward's palace. Elphir claimed he had drawn the short straw, being expected to sleep in Denethor's bed, Amrothos had been hardly more comfortable with the prospect of entering Boromir's chamber, kept untouched, almost shrine-like, by Boromir's father. They both said he was the lucky one, getting Faramir's chamber, but in truth Erchirion found the thought of his cousin lying on the brink of death, burning with fever, was almost beyond bearing.

Nonetheless, Erchirion was so exhausted he barely had a chance to shed his armour and outer garments, then pitch into the bed and pull the coverlet up before he passed into a dreamless sleep.

Judging from the gritty feeling behind his eyes, he woke barely three hours later, and knew instantly that he would not go back to sleep. Imrahil had said he would send servants to rouse his sons two hours before the dawn, anticipating that the enemy would attack an hour later, still under cover of darkness, at that point when those defending the city were at their weakest and most disorientated. If Erchirion judged right (and nearly a decade as a soldier had given him a knack for judging time) there was still at least an hour to go before the servants arrived.

_Might as well stay in the warm_ , he thought to himself. _Even if I can't sleep, at least I'll be reasonably rested, not stiff and chilled to the bone from standing wound tight as a drum skin upon the ramparts in the dark._

He swung his legs over the edge of the bed, took the candle holder from the bedside, and in the dim glow of the embers, made his way to the fireplace where he kindled the candle. He made his way over to Faramir's desk, looking for something to read, to distract him for the next hour, and lit the candles in the candelabra beside the stacks of papers. Half-a-dozen of them! Faramir was obviously accustomed to work long after the sun had set, and made sure his desk was well lit.

Erchirion sat down, and started to browse through the books and papers lying across its surface. Then a couple in particular caught his eye. He pulled them out of the old leather folder in which they lay and looked at them. Charcoal sketches, one in a slightly wobbly hand, of a crudely drawn set of arcs, then a second sketch in a rather firmer hand, the arcs more neatly drawn and more varied in height. His and Elphir's.

A wave of sadness hit him once more. The memory of that sunny day down at the lists came flooding back to him. He had idolised Faramir as a child. Both his cousins had been attentive to the Dol Amroth children, but Faramir in particular seemed to feel a special care towards them, entering into their games and pastimes with affectionate enthusiasm. And now Boromir was dead and Faramir dying.

Erchirion gave his head a shake. Maudlin thoughts would not help him. He pulled out the sheaf of papers that went with the sketches and began to read.

_**On the Geometrical Relationships underlying the Science of Ballistics** _

The heading was neatly underlined. Had his cousin been writing a book, an academic treatise of some kind?

_Iorhael explains motion to be of two types: firstly the motion of an object towards its natural place, for instance, when a stone is dropped towards the earth of which it is made; and secondly the motion of an object, animal or person kept in a state of motion by repeated application of an impulse, for example the moving hooves or feet of a running horse or man._

Despite the sadness thoughts of his cousin occasioned, Erchirion smiled. How like Faramir. To begin with a scholarly summation of what others knew of a matter, before adding one's own thoughts.

_In this, Iorhael is for the most part correct, but extensive experimentation shows the matter to be more complex. It is necessary to distinguish between motion at a right angle to the surface of the earth, which may indeed be conceived of as the tendency of an object to seek or return to its natural place, and that parallel to the surface of the earth. The laws governing these two limiting cases are slightly different. An arrow shot straight upwards will slow, stop, then descend. An arrow shot along a flat path (provided that path is not of excessive length) will continue at a constant speed, as may be ascertained testing this hypothesis by measuring the distance an arrow penetrates into a target when loosed from different distances._

Erchirion had a sudden mental image of his cousin at the butts, carefully loosing arrow after arrow into targets at different distances, controlling his draw to loose them one after another with identical force, with a precision born of years and years of practice, then carefully measuring how deeply they had penetrated into the woven straw targets. He could picture the look of concentration, and absorbed interest as the task was carried out. Only Faramir, he thought to himself.

_The upwards and downwards motions, in contrast, consist of variations in speed as the arrow reaches its zenith then falls to its nadir. The height, and hence time this motion takes depends on the force with which the arrow is loosed. Bows with a low draw-weight will not project an arrow as far into the air as those from bows with a high draw-weight (as has been tested exhaustively using arrows with lengths of cord attached to measure the height attained)._

_We may apply this to the question of launching missiles from a ballista thus: the angle to which the ballista is set will determine what proportion of the motion proceeds horizontally, and is thus subject to the first law, and what proceeds vertically, and is thus subject to the second. It is the second which determines the time of flight, and hence how far the missile will travel. Too steep an angle will result in a flight of long duration, but without much forwards motion, so that the missile will fall short of its target; too shallow an angle and although the forwards motion be greater, the missile will not remain in flight for sufficient time to reach its target. Thus the optimum angle to fire the missile consists of half of a right angle._

Erchirion read these notes with fascination; here was the reasoning behind something the ballista crews upon the walls knew from experience: that one launched missiles at this angle if one wanted them to go the greatest distance. But seeing it written down this way gave him an extra clue as to how to more efficiently effect straddles – the trial and error method used to bracket a target and narrow down the precise range. One could, he realised, disregard all launch angles shallower than half a right angle. He paused for long enough to see one final sheet of parchment, a set of scratched and crossed-out calculations and half-finished attempts at geometrical proofs, with, beneath it all, a few terse, frustrated sentences.

_Buggeration. There must be a geometrical relationship here, but I cannot for the life of me see what it is. The height grows at a slower rate with elapsing time – as the square? How could this be tested?_

Again, so like Faramir. He so very rarely lost his composure, but on the handful of occasions when he did, could swear like a sailor. Erchirion had no doubt that, given time and leisure, the geometrical proof would have come to Faramir. He had always been singularly gifted with such things. But alas, poor cousin. He would now die, burned from within, without completing his proof.

Then suddenly the practical implications of what he had been reading struck Erchirion. He was shaken from his maudlin stupor in an instant. Without wasting a moment more, he pulled on his outer garments and fixed what he could of his armour, then went in search of a squire to help him with the rest of the buckles. That done, he strode off down the narrow streets towards the outermost wall, in search of Prince Imrahil.

“Father,” he said, clasping his hand to his heart and bowing his head.

“Erchirion! I thought I told you and your brothers to rest, in preparation for this morning's anticipated assault.”

“And I have, Sir. But I have been looking through some of Faramir's papers, and I think I have found a tactic we could usefully employ. We need to be able to launch a counter attack, covered by our own archers upon the ramparts, do we not?”

“Yes. The Swan Knights will ride forth.”

“And the biggest threat to this is the enemy's siege towers, which will be able to rain arrows down not just upon our knights, but upon our archers high on the ramparts – for their towers are higher than our battlements.”

“That,” said Imrahil, with the weary tone of one explaining the basics of siege warfare to one's newest, most junior officer, “Is something of the point of siege towers, yes.”

“But we have ballistas. So far we have simply been using them to throw the enemy's troops into terror and confusion. But I think Faramir's papers may have just given me the method for range-finding, quickly and accurately. If we can set fire to the siege towers with barrels of burning pitch, I think it may adjust the odds slightly towards us.” (Such was Erchirion's intrinsic honesty that he could not quite bring himself to go so far as to say “in our favour.”)

Imrahil looked at him. “If you think you can do this, then hasten to the ramparts and see what can be accomplished.” 

Erchirion bowed once more, then turned and made for the nearest of the stairs that led to the ramparts.

_Thirteen hours later_

Erchirion sat beside Faramir's bed. Faramir lay in a deep sleep, but his face was now its normal colour, and his hair now spread across the pillow rather than being plastered to a sweating brow. His chest rose and fell gently, his breathing even and unlaboured. Erchirion, for his part, had shed his armour, and sat relatively at ease in breeches and a loose tunic. He pushed his hair back from his face (the leather thong which had bound it now lay somewhere trampled into the mud on the Pelennor fields), then reached out towards the patient.

“You can't hear me, cousin, but you played a part today, even as you lay upon your sick bed.” The young prince found his words rambling slightly as he patted the older man's hand. “You have no idea how relieved I am to be able to say merely 'sick bed', and to mean it. Anyway, now you will live to finish your geometrical proofs. And they have already met their first test today, and passed it with flying colours. Once I realised that there was no point in using angles less than half a right angle, I was able to range-find in half the time it would previously have taken, so we could destroy twice as many of the enemy's siege towers. Perhaps a small part of the battle… Just a tiny corner. But maybe it played its role in buying us the time we needed until the Rohirrim arrived.”

“Erchi, come and eat!” It was Amrothos, standing in the doorway. Erchirion gave Faramir's hand one last squeeze, then followed his brother. As he walked through the quiet passageways of the Houses of Healing, he found himself musing on relationships of blood, and relationships of numbers, and the unexpected and blessed consequences of the combination of the two.


End file.
